This is a crosspost of my question from MSE from 3 weeks ago which was bountied but has received no response.

For an algebra assignment, I was asked to do a literature review and write up a proof of the transcendence of $\pi$. All sources I found presented Lindemann's proof (the "Transcendence of $\pi$" by Steve Mayer is an example), which uses symmetric function theory (particularly, the fundamental theorem of elementary symmetric functions) as a critical component of its argument, and as far as I am aware, generalizations like the Lindemann-Weierstrass theorem are similarly reliant on symmetric function theory in their proofs. Before the assignment, I did not know any symmetric function theory (perhaps this is a fault of my undergraduate education), and I found it the hardest part of the proof to learn and understand.

Now, learning the basics of symmetric function theory was worthwhile and will probably be useful for me in the future, but it begs the question: Is there a proof of the transcendence of $\pi$ without symmetric function theory, and if not, why?

As brought up by Paramanand Singh in the MSE thread, a modification of Lindemann's argument that proves the same claim in the argument where symmetric function theory was used originally (a certain polynomial constructed in the proof has integer coefficients) would suffice, though I'd also accept a "higher-tech" proof that is a different argument entirely, if you know of any that exist.

Algebra), does not use symmetric functions. It's much less elementary than the usual expositions of Hermite-Lindemann's proofs, but on the other hand, it feels more natural and less ad hoc (to me). $\endgroup$